9x9 Queens Puzzle #106

About 9x9 Queens puzzle #106

Board #106 of the 9x9 bank carries a medium rating. Its 81 squares are shared between nine colored regions, and the solution places nine queens: one per row, one per column, one per region, none of them side by side or corner to corner.

As with every puzzle in this collection, #106 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.

How the colors divide this board

From smallest to largest, the regions on puzzle #106 are: indigo (1 square), amber (1 square), orange (1 square), khaki (5 squares), slate (8 squares), mint (11 squares), blue (14 squares), red (20 squares), pale yellow (20 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 20 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; pale yellow has 20. The spread between them on this board is 19 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while red is the most scattered at 56% of its 6×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

Region by region

Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 6–6). It never leaves row 1, so its queen is locked to that row.

Amber occupies 1 square within rows 2–2, columns 1–1. It never leaves row 2, so its queen is locked to that row.

Orange: 1 square across rows 9–9, columns 9–9. It never leaves row 9, so its queen is locked to that row.

Khaki covers 5 squares, spanning 2 rows and 3 columns (rows 8–9, columns 5–7).

Slate occupies 8 squares within rows 2–5, columns 1–3.

Mint: 11 squares across rows 1–4, columns 7–9.

Blue covers 14 squares, spanning 4 rows and 6 columns (rows 1–4, columns 1–6).

Red occupies 20 squares within rows 4–9, columns 4–9.

Pale yellow: 20 squares across rows 4–9, columns 1–5.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 6 and 6. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #106.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.

Work outward to the other flat regions next. Amber, orange and khaki are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.

The tightest lines on puzzle #106

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 7 are the narrowest, touched by only 2 colors. Column 8 is the tightest column, with 2 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 8 rows. Row 4 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.

The whole middle game reduces to two sweeps. Sweep the nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 solve from a stalled one.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #106 is rated medium

The medium label comes out of four measurable things about #106: six of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 19. Boards that score tight on those measures give up their first queen fast.

Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.

A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.

Techniques that work well at 9x9

At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #106 the place to look for one is around row 6.

Use this as a rough check on your progress: the completed board has 3 queens around the border and 6 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.